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Original Articles

Further research on limit theorems for self-normalized sums

Pages 385-402 | Received 26 Jan 2018, Accepted 18 Oct 2018, Published online: 28 Dec 2018

References

  • Bentkus, V., and F. Götze. 1996. The Berry-Esseen bound for student’s statistic. The Annals of Probability 24 (1):491–503. doi:10.1214/aop/1042644728.
  • Berkes, I., and E. Csáki. 2001. A universal result in almost sure Central limit theory. Stochastic Processes and Their Applications 94 (1):105–34. doi:10.1016/S0304-4149(01)00078-3.
  • Billingsley, P. 1968. Convergence of probability measures. New York: Wiley.
  • Brosamler, G. A. 1988. An almost everywhere Central limit theorem. Mathematical Proceedings of the Cambridge Philosophical Society 104 (3):561–74. doi:10.1017/S0305004100065750.
  • Csörgő, M., B. Szyszkowicz, and Q. Y. Wang. 2003a. Darling-Erdös theorem for self-normalized partial sums. The Annals of Probability 31 (2):676–92. doi:10.1214/aop/1048516532.
  • Csörgő, M., B. Szyszkowicz, and Q. Y. Wang. 2003b. Donsker’s theorem for self-normalized partial sums processes. The Annals of Probability 31 (3):1228–40. doi:10.1214/aop/1055425777.
  • De la Peña, V. H., M. J. Klass, and T. L. Lai. 2009. Self-normalized processes- Limit theory and statistical applications. Probability and its applications. New York; Berlin: Springer-Verlag.
  • Feng, F. X., D. C. Wang, and Q. Y. Wu. 2016. An almost sure Central limit theorem for self-normalized weighted sums of the ϕ mixing random variables. Journal of Mathematical Inequalities 10 (1):233–45. doi:10.7153/jmi-10-20.
  • Giné, E., F. Götze, and D. M. Mason. 1997. When is the student t-statistic asymptotically standard normal? The Annals of Probability 25 (3):1514–31. doi:10.1214/aop/1024404523.
  • Gonchigdanzan, K., and G. A. Rempała. 2006. A note on the almost sure limit theorem for the product of partial sums. Applied Mathematics Letters 19 (2):191–6. doi:10.1016/j.aml.2005.06.002.
  • Griffin, P. S., and J. D. Kuelbs. 1989. Self-normalized laws of the iterated logarithm. The Annals of Probability 17 (4):1571–601. www.jstor.org/stable/2244456.
  • Hörmann. S. 2006. An extension of almost sure Central limit theory. Statistics & Probability Letters 76 (2):191–202. doi: 10.1016/j.spl.2005.07.015.
  • Huang, S. H., and T. X. Pang. 2010. An almost sure Central limit theorem for self-normalized partial sums. Computers & Mathematics with Applications 60 (9):2639–44. doi:10.1016/j.camwa.2010.08.093.
  • Li, Y. X. 2013. An extension of the almost sure Central limit theorem for products of sums under association. Communications in Statistics-Theory and Methods 42 (3):478–90. doi:10.1080/03610926.2011.581790.
  • Lin, Z. Y. 1996. A self-normalized Chung-type law of iterated logarithm. Theory of Probability and Its Applications 41 (4):791–8. doi:10.1137/TPRBAU000041000004000741000001.
  • Lin, Z. Y., and C. R. Lu. 1996. Limit theory for mixing dependent random variables. Beijing: Science Press.
  • Lin, Z. Y., T. X. Pang, and K. S. Hwang. 2016. An almost sure Central limit theorem for self-normalized partial sums of weakly dependent random variables. Communications in Statistics-Theory and Methods 45 (12):3411–20. doi:10.1080/03610926.2013.776688.
  • Miao, Y. 2009. An extension of almost sure Central limit theory for the product of partial sums. Journal of Dynamical Systems and Geometric Theories 7 (1):49–60. doi:10.1080/1726037X.2009.10698562.
  • Peligrad, M., and P. Reévész. 1989. On the almost sure central limit theorem. In Almost everywhere convergence. vol. II, 209–25. Boston, MA: Academic Press.
  • Peligrad, M., and Q. M. Shao. 1995. A note on the almost sure Central limit theorem. Statistics & Probability Letters 22 (2):131–6. doi:10.1016/0167-7152(94)00059-H.
  • Račkauskas, A., and C. Suquet. 2011. Functional Central limit theorems for self-normalized partial sums of linear processes. Lithuanian Mathematical Journal 51 (2):251–9. doi:10.1007/s10986-011-9123-7.
  • Schatte, P. 1988. On strong versions of the Central limit theorem. Mathematische Nachrichten 137 (1):249–56. doi:10.1002/mana.19881370117.
  • Shao, Q. M. 1997. Self-normalized large deviations. The Annals of Probability 25 (1):285–328. doi:10.1214/aop/1024404289.
  • Shao, Q. M. 1999. A Cramér type large deviation result for student’s t-statistics. Journal of Theoretical Probability 12 (2):385–98. doi:10.1023/A:1021626127372.
  • Shao, Q. M., and Q. Y. Wang. 2013. Self-normalized limit theorems: A survey. Probability Surveys 10:69–93. doi:10.1214/13-PS216.
  • Tan, Z. Q. 2015. Almost sure Central limit theorem for exceedance point processes of stationary sequences. Brazilian Journal of Probability and Statistics 29 (3):717–31. doi:10.1214/14-BJPS242.
  • Tong, B., Z. X. Peng, and N. Saralees. 2009. An extension of almost sure Central limit theorem for order statistics. Extremes 12 (3):201–9. doi:10.1007/s10687-008-0075-1.
  • Wu, Q. Y. 2015. Improved results in almost sure Central limit theorems for the maxima and partial sums for Gaussian sequences. Journal of Inequalities and Applications 2015 (109):15. doi: 10.1186/s13660-015-0634-3.
  • Zhang, Y., and X. Y. Yang. 2011. An almost sure Central limit theorem for self-normalized products of sums of i.i.d. random variables. Journal of Mathematical Analysis and Applications 376 (1):29–41. doi:10.1016/j.jmaa.2010.10.021.
  • Zhang, Y., and X. Y. Yang. 2013. An almost sure Central limit theorem for self-normalized weighted sums. Acta Mathematicae Applicatae Sinica, English Series 29 (1):79–92. doi:10.1007/s10255-010-8247-6.
  • Zhang, Y. 2016. A universal result in almost sure Central limit theorem for products of sums of partial sums under mixing sequence. Stochastics 88 (6):803–12. doi:10.1080/17442508.2016.1146281.
  • Zhang, Y. 2016. An extension of almost sure Central limit theorem for self-normalized products of sums for mixing sequences. Communications in Statistics-Theory and Methods 45 (22):6625–40. doi:10.1080/03610926.2014.963619.

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